advertisement

Interpreting the RGA Andrew Kim Stephanie Cleto What is the RGA? • Relative Gain Array is an analytical tool used to determine the optimal input-output variable pairings for a multi-input-multi-output (MIMO) system. Significance of the RGA • Below is a general relative gain array Significance of the RGA • Below is a general relative gain array • Different columns represent the different manipulated (input) variables Significance of the RGA • Below is a general relative gain array • Different columns represent the different manipulated (input) variables • Different rows represent the different controlled (output) variables Significance of the RGA • The values in the array describe the relationship between the input and output variables • Negative values indicate an unstable relationship • A value of 0 indicates no relationship • A value of 1 indicates that specific input variable is the only influence on that output variable Example 1 • Assume a mixing tank with constant mass and two inputs as shown below: • wA and wB are manipulated flowrates entering the tank • w is the flowrate leaving the tank and xA is the concentration of A in the tank Example from http://eweb.chemeng.ed.ac.uk/courses/control/restricted/course/advanced/casestudy/exercise2.html Example 1 (cont.) • This process can be modeled by the following equations: w = wA + wB xA = wA/(wA + wB) • The RGA can be solved for this system: 1 x x RelativeGain Array 1 x x Example 1 Solution • What does the RGA tell us? – If a concentration of xA=0.5 is desired, either wA or wB can be used – If a concentration of xA>0.5 is desired, then the concentration loop should be paired with wA – If a concentration of xA<0.5 is desired, then the concentration loop should be paired with wB Example 2 Run R(kg/min) S(kg/min) xD xB 1 125 22 0.97 0.04 2 150 22 0.93 0.06 3 150 20 0.91 0.08 Adapted from http://eweb.chemeng.ed.ac.uk/courses/control/restricted/course/advanced/casestudy/exercise2.html Example 2 Solution • RGA Matrix 2 -1 -1 2 • Pairing of variables: RGA matrix value should be 1)positive, then 2)close to 1. • Because each combination has only one positive value, that is the combination to be paired (R,xD and P,xB) Example 3 • Suppose you calculate the following RGA matrix. How should pairing of the variables occur? -.25 0 1.25 0.75 0.8 -0.55 0.5 0.2 0.3 Example 3 Solution • In the first row, only x3 gives a positive result, and then we go with the closest values to 1 for the others. • y1,x3 y2,x2 y3,x1 -.25 0 1.25 0.75 0.8 -0.55 0.5 0.2 0.3